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4 THE PRINCIPLE OF ANALYTIC CONTINUATION 209

(9.4.4) Let A c C^p be an open connected set, f and g two analytic functions
in A with values in a complex Banach space E. Let U be an open subset of
A, b a point of U, and suppose that f(x) ~g(x) in the set U n (b + R^p);
thenf(x) = g(x)for every x e A.

We can suppose, by a translation, that b = 0; let h=f—g_y and let P
be a polydisk in (7, of center 0, contained in U and such that in P, h(z)
is equal to the sum of an absolutely summable power series (c_vz^v). Now
P n R^p is a polydisk in R^p, and h(x) = 0 in P n W; this shows by (9.1.6)
that c_v = 0 for every v, hence h(z) = 0 in P and (9.4.2) can be applied.

In an open connected set A c K^p, we say that a subset M c A is a set
of uniqueness if any two functions, defined and analytic in A, coincide in A
as soon as they coincide in M. (9.4.2), (9.4.3), and (9.4.4) show that a non-
empty open subset U of A, or the intersection U n (b + R^p) (if not empty),
or, for/? = 1, a compact infinite subset of A, are sets of uniqueness. We shall
see another example in Section 9.9 for K = C.

The preceding result shows that if an open connected subset A c= C^pis such that A n R^p ^ 0, any analytic function / in A is completely deter-
mined by its values in A n W. The restriction of/to A n R^p is an analytic
function, but in general an analytic function in A n W cannot be extended
to an analytic function in A; we have however the weaker result:

(9.4.5) Let Ebea complex Banach space, A an open subset of W, fan analytic
mapping of A into E. Then there is an open set B c C^p such that B n R^p = A,
and an analytic mapping gofE into E which extends f.

Indeed, for each a = (a_i9..., a_p) e A, there is an open polydisk P_a in
W defined by \x_i — a_i\<r_i (l^i^p) contained in A and such that,
in P_a, f(x) is equal to the sum of an absolutely summable power
series (c_ni ...„/*! - fli)"¹... (^p-a_p)"*)- Let Q_fl be the open polydisk
in C^p, of center a and radii r_t; then, by (9.1.2), the power series
fan...n_p(^zi - tfi)^Wl • • • (z_p - a_PY^p) ^ absolutely summable in Q_a; let g_a(z) be
its sum. If a, b are two points of A such that Q_fl n Q_b ^ 0, then P_fl n P₆ =
(Pa n Q_b) n R^p is not empty, and we have g_a(x) = g_b(x) =f(x) in P_a n P₆.
Moreover Q_a n Q_fc is connected by (9.1.1); it follows from (9.4.4) that
g_a(z) = g_b(z) in Q_fl n Q₆. We can now take B = (J Q_a, and define g as

aeA

equal to#_fl in each Q_fl; the analyticity of g follows from (9.3.1).

The proof of (9.4.5) shows that when/is an entire function defined in R^p,
it can be extended to an entire function defined in C^p, and that function is
unique by (9.4.4).